To solve this, let's compute the left endpoint Riemann sum for the function $f(x) = \frac{x^2}{11}$ on the interval $[2, 6]$, divided into equal subintervals.

Step 1: Divide the interval

• The interval $[2, 6]$ has 4 subintervals.
• Subinterval width ($\Delta x$) is: $\Delta x = \frac{6 - 2}{4} = 1$

Step 2: Identify left endpoints

The left endpoints of the subintervals are: $x = 2, 3, 4, 5$

Step 3: Evaluate $f(x)$ at each left endpoint

Substitute each left endpoint into $f(x) = \frac{x^2}{11}$:

• $f(2) = \frac{2^2}{11} = \frac{4}{11}$
• $f(3) = \frac{3^2}{11} = \frac{9}{11}$
• $f(4) = \frac{4^2}{11} = \frac{16}{11}$
• $f(5) = \frac{5^2}{11} = \frac{25}{11}$

Step 4: Compute the Riemann sum

The left endpoint Riemann sum is: $\text{Sum} = \Delta x \cdot \left[ f(2) + f(3) + f(4) + f(5) \right]$ Substitute values: $\text{Sum} = 1 \cdot \left( \frac{4}{11} + \frac{9}{11} + \frac{16}{11} + \frac{25}{11} \right)$ Combine terms: $\text{Sum} = \frac{4 + 9 + 16 + 25}{11} = \frac{54}{11}$

Thus, the value of the left endpoint Riemann sum is: $\boxed{\frac{54}{11} \, \text{or approximately } 4.91}$

Let me know if you'd like further clarification or details!
Here are related questions you might explore:

1. How do you compute right endpoint Riemann sums for a function?
2. What happens to the Riemann sum as the number of subintervals increases?
3. Can this Riemann sum approximate the exact integral of $f(x)$?
4. How does the choice of endpoints affect the sum?
5. How do Riemann sums relate to definite integrals in calculus?

Tip: For better approximations, try using midpoint Riemann sums or evaluate the definite integral analytically.